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# Rational numbers (Class 8th)  - Revision Notes

## Rational Numbers

A number is called Rational if it can be expressed in the form p/q where p and q are integers (q > 0). It includes all natural, whole number and integers.

Example: 1/2, 4/3, 5/7,1 etc.

## Natural Numbers

All the positive integers from 1, 2, 3,……, ∞.

### Whole Numbers

All the natural numbers including zero are called Whole Numbers.

Integers

All negative and positive numbers including zero are called Integers.

## Properties of Rational Numbers

### 1. Closure Property

This shows that the operation of any two same types of numbers is also the same type or not.

a. Whole Numbers

If p and q are two whole numbers then

 Operation Addition Subtraction Multiplication Division Whole number p + q will also be the whole number. p – q will not always be a whole number. pq will also be the whole number. p ÷ q will not always be a whole number. Example 6 + 0 = 6 8 – 10 = – 2 3 × 5 = 15 3 ÷ 5 = 3/5 Closed or Not Closed Not closed Closed Not closed

b. Integers

If p and q are two integers then

 Operation Addition Subtraction Multiplication Division Integers p+q will also be an integer. p-q will also be an integer. pq will also be an integer. p ÷ q will not always be an integer. Example - 3 + 2 = – 1 5 – 7 = – 2 - 5 × 8 = – 40 - 5 ÷ 7  = – 5/7 Closed or not Closed Closed Closed Not  closed

c. Rational Numbers

If p and q are two rational numbers then

 Operation Addition Subtraction Multiplication Division Rational Numbers p + q will also be a rational number. p – q will also be a rational number. pq will also be a rational number. p ÷ q will not always be a rational number Example p ÷ 0 = not defined Closed or Not Closed Closed Closed Not closed

### 2. Commutative Property

This shows that the position of numbers does not matter i.e. if you swap the positions of the numbers then also the result will be the same.

a. Whole Numbers

If p and q are two whole numbers then

 Operation Addition Subtraction Multiplication Division Whole number p + q = q + p p – q ≠ q – p p × q = q × p p ÷ q ≠ q ÷ p Example 3 + 2 = 2 + 3 8 –10 ≠ 10 – 8 – 2 ≠ 2 3 × 5 = 5 × 3 3 ÷ 5 ≠ 5 ÷ 3 Commutative yes No yes No

b. Integers

If p and q are two integers then

 Operation Addition Subtraction Multiplication Division Integers p + q = q + p p – q ≠ q – p p × q = q × p p ÷ q ≠ q ÷ p Example True 5 – 7 = – 7 – (5) - 5 × 8 = 8 × (–5) - 5 ÷ 7 ≠ 7 ÷ (-5) Commutative yes No yes No

c. Rational Numbers

If p and q are two rational numbers then

 Operation Addition Subtraction Multiplication Division Rational numbers p + q = q + p p –q ≠ q - p p × q = q × p p ÷ q ≠ q ÷ p Example Commutative yes No yes No

### 3. Associative Property

This shows that the grouping of numbers does not matter i.e. we can use operations on any two numbers first and the result will be the same.

a. Whole Numbers

If p, q and r are three whole numbers then

 Operation Addition Subtraction Multiplication Division Whole number p + (q + r) = (p + q) + r p – (q – r) = (p – q) – r p × (q × r) = (p × q) × r p ÷ (q ÷ r)  ≠ (p ÷ q) ÷ r Example 3 + (2 + 5) = (3 + 2) + 5 8 – (10 – 2) ≠ (8 -10) – 2 3 × (5 × 2) = (3 × 5) × 2 10 ÷ (5 ÷ 1) ≠ (10 ÷ 5) ÷ 1 Associative yes No yes No

b. Integers

If p, q and r are three integers then

 Operation Integers Example Associative Addition p + (q + r) = (p + q) + r (– 6) + [(– 4)+(–5)] = [(– 6) +(– 4)] + (–5) Yes Subtraction p – (q – r) = (p – q) – r 5 – (7 – 3) ≠ (5 – 7) – 3 No Multiplication p × (q × r) = (p × q) × r (– 4) × [(– 8) ×(–5)] = [(– 4) × (– 8)] × (–5) Yes Division p ÷ (q ÷ r) ≠ (p ÷ q) ÷ r [(–10) ÷ 2] ÷ (–5) ≠ (–10) ÷ [2 ÷ (– 5)] No

c. Rational Numbers

If p, q and r are three rational numbers then

 Operation Integers Example Associative Addition p + (q + r) = (p + q) + r yes Subtraction p – (q – r) = (p – q) – r No Multiplication p × (q × r) = (p × q) × r yes Division p ÷ (q ÷ r)  ≠ (p ÷ q) ÷ r No

## The Role of Zero in Numbers (Additive Identity)

Zero is the additive identity for whole numbers, integers and rational numbers.

 Identity Example Whole number a + 0 = 0 + a = a Addition of zero to whole number 2 + 0 = 0 + 2 = 2 Integer b + 0 = 0 + b = b Addition of zero to an integer False Rational number c + 0 = 0 + c = c Addition of zero to a rational number 2/5 + 0 = 0 + 2/5 = 2/5

## The Role of one in Numbers (Multiplicative Identity)

One is the multiplicative identity for whole numbers, integers and rational numbers.

 Identity Example Whole number a ×1 = a Multiplication of one to the whole number 5 × 1 = 5 Integer b × 1= b Multiplication of one to an integer - 5 × 1 = – 5 Rational Number c × 1= c Multiplication of one to a rational number

## Negative of a Number (Additive Inverse)

 Identity Example Whole number a +(- a) = 0 Where a is a  whole number 5 + (-5) = 0 Integer b +(- b) = 0 Where b is an integer True Rational number c + (-c) = 0 Where c is a rational number

## Reciprocal (Multiplicative Inverse)

The multiplicative inverse of any rational number

Example

The reciprocal of 4/5 is 5/4.

### Distributivity of Multiplication over Addition and Subtraction for Rational Numbers

This shows that for all rational numbers p, q and r

1. p(q + r) = pq + pr

2. p(q – r) = pq – pr

Example

Check the distributive property of the three rational numbers 4/7,-( 2)/3 and 1/2.

Solution

Let’s find the value of

This shows that

### Representation of Rational Numbers on the Number Line

On the number line, we can represent the Natural numbers, whole numbers and integers as follows

Rational Numbers can be represented as follows

## Rational Numbers between Two Rational Numbers

There could be n number of rational numbers between two rational numbers. There are two methods to find rational numbers between two rational numbers.

Method 1

We have to find the equivalent fraction of the given rational numbers and write the rational numbers which come in between these numbers. These numbers are the required rational numbers.

Example

Find the rational number between 1/10 and 2/10.

Solution

As we can see that there are no visible rational numbers between these two numbers. So we need to write the equivalent fraction.

2/10 = 20/100((multiply the numerator and denominator by 10)

Hence, 2/100, 3/100, 4/100……19/100 are all the rational numbers between 1/10 and 2/10.

Method 2

We have to find the mean (average) of the two given rational numbers and the mean is the required rational number.

Example

Find the rational number between 1/10 and 2/10.

Solution

To find mean we have to divide the sum of two rational numbers by 2.

3/20 is the required rational numbers and we can find more by continuing the same process with the old and the new rational number.

Remark: 1. This shows that if p and q are two rational numbers then (p + q)/2 is a rational number between p and q so that

p < (p + q)/2 < q.

2. There are infinite rational numbers between any two rational numbers.